Singularities on demi-normal varieties

Abstract

The birational classification of varieties inevitably leads to the study of singularities. The types of singularities that occur in this context have been studied by Mori, Kollar, Reid, and others, beginning in the 1980s with the introduction of the Minimal Model Program. Normal singularities that are terminal, canonical, log terminal, and log canonical, and their non-normal counterparts, are typically studied by using a resolution of singularities (or a semi-resolution), and finding numerical conditions that relate the canonical class of the variety to that of its resolution. In order to do this, it has been assumed that a variety X is has a $${\mathbb {Q}}$$ -Cartier canonical class: some multiple $$mK_X$$ of the canonical class is Cartier. In particular, this divisor can be pulled back under a resolution $$f: Y \rightarrow X$$ by pulling back its local sections. Then one has a relation $$K_Y \sim \frac{1}{m}f^*(mK_X) + \sum a_iE_i$$ . It is then the coefficients of the exceptional divisors $$E_i$$ that determine the type of singularities that belong to X. It might be asked whether this $${\mathbb {Q}}$$ -Cartier hypothesis is necessary in studying singularities in birational classification. de Fernex and Hacon (Compos Math 145:393–414, 2009) construct a boundary divisor $$\Delta $$ for arbitrary normal varieties, the resulting divisor $$K_X + \Delta $$ being $${\mathbb {Q}}$$ -Cartier even though $$K_X$$ itself is not. This they call (for reasons that will be made clear) an m-compatible boundary for X, and they proceed to show that the singularities defined in terms of the pair $$(X,\Delta )$$ are none other than the singularities just described, when $$K_X$$ happens to be $${\mathbb {Q}}$$ -Cartier. Thus, a wider context exists within which one can study singularities of the above types. In the present paper, we extend the results of de Fernex and Hacon (Compos Math 145:393–414, 2009) still further, to include demi-normal varieties without a $${\mathbb {Q}}$$ -Cartier canonical class. Our main result is that m-compatible boundaries exist for demi-normal varieties (Theorem 1.1). This theorem provides a link between the theory of singularities for arbitrary demi-normal varieites (whose canonical class may not $$\hbox {be } {\mathbb {Q}}$$ -Cartier), that theory being developed in the present paper, and the established theory of singularities of pairs.